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Theorem mon1pval 19527
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p  |-  P  =  (Poly1 `  R )
uc1pval.b  |-  B  =  ( Base `  P
)
uc1pval.z  |-  .0.  =  ( 0g `  P )
uc1pval.d  |-  D  =  ( deg1  `  R )
mon1pval.m  |-  M  =  (Monic1p `  R )
mon1pval.o  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mon1pval  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Distinct variable groups:    B, f    D, f    .1. , f    R, f    .0. , f
Allowed substitution hints:    P( f)    M( f)

Proof of Theorem mon1pval
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 mon1pval.m . 2  |-  M  =  (Monic1p `  R )
2 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  (Poly1 `  r )  =  (Poly1 `  R ) )
3 uc1pval.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
42, 3syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  (Poly1 `  r )  =  P )
54fveq2d 5529 . . . . . 6  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  ( Base `  P
) )
6 uc1pval.b . . . . . 6  |-  B  =  ( Base `  P
)
75, 6syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ( Base `  (Poly1 `  r ) )  =  B )
84fveq2d 5529 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  ( 0g `  P ) )
9 uc1pval.z . . . . . . . 8  |-  .0.  =  ( 0g `  P )
108, 9syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  ( 0g `  (Poly1 `  r ) )  =  .0.  )
1110neeq2d 2460 . . . . . 6  |-  ( r  =  R  ->  (
f  =/=  ( 0g
`  (Poly1 `  r ) )  <-> 
f  =/=  .0.  )
)
12 fveq2 5525 . . . . . . . . . 10  |-  ( r  =  R  ->  ( deg1  `  r )  =  ( deg1  `  R ) )
13 uc1pval.d . . . . . . . . . 10  |-  D  =  ( deg1  `  R )
1412, 13syl6eqr 2333 . . . . . . . . 9  |-  ( r  =  R  ->  ( deg1  `  r )  =  D )
1514fveq1d 5527 . . . . . . . 8  |-  ( r  =  R  ->  (
( deg1  `
 r ) `  f )  =  ( D `  f ) )
1615fveq2d 5529 . . . . . . 7  |-  ( r  =  R  ->  (
(coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( (coe1 `  f ) `  ( D `  f ) ) )
17 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
18 mon1pval.o . . . . . . . 8  |-  .1.  =  ( 1r `  R )
1917, 18syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
2016, 19eqeq12d 2297 . . . . . 6  |-  ( r  =  R  ->  (
( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  =  ( 1r `  r
)  <->  ( (coe1 `  f
) `  ( D `  f ) )  =  .1.  ) )
2111, 20anbi12d 691 . . . . 5  |-  ( r  =  R  ->  (
( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) )  <->  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  )
) )
227, 21rabeqbidv 2783 . . . 4  |-  ( r  =  R  ->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) }  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
23 df-mon1 19516 . . . 4  |- Monic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  =  ( 1r `  r ) ) } )
24 fvex 5539 . . . . . 6  |-  ( Base `  P )  e.  _V
256, 24eqeltri 2353 . . . . 5  |-  B  e. 
_V
2625rabex 4165 . . . 4  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  e.  _V
2722, 23, 26fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
28 fvprc 5519 . . . 4  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  (/) )
29 ssrab2 3258 . . . . . 6  |-  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  B
30 fvprc 5519 . . . . . . . . . 10  |-  ( -.  R  e.  _V  ->  (Poly1 `  R )  =  (/) )
313, 30syl5eq 2327 . . . . . . . . 9  |-  ( -.  R  e.  _V  ->  P  =  (/) )
3231fveq2d 5529 . . . . . . . 8  |-  ( -.  R  e.  _V  ->  (
Base `  P )  =  ( Base `  (/) ) )
336, 32syl5eq 2327 . . . . . . 7  |-  ( -.  R  e.  _V  ->  B  =  ( Base `  (/) ) )
34 base0 13185 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
3533, 34syl6eqr 2333 . . . . . 6  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3629, 35syl5sseq 3226 . . . . 5  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/) )
37 ss0 3485 . . . . 5  |-  ( { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  C_  (/)  ->  { f  e.  B  |  (
f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3836, 37syl 15 . . . 4  |-  ( -.  R  e.  _V  ->  { f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }  =  (/) )
3928, 38eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  (Monic1p `  R )  =  {
f  e.  B  | 
( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) } )
4027, 39pm2.61i 156 . 2  |-  (Monic1p `  R
)  =  { f  e.  B  |  ( f  =/=  .0.  /\  ( (coe1 `  f ) `  ( D `  f ) )  =  .1.  ) }
411, 40eqtri 2303 1  |-  M  =  { f  e.  B  |  ( f  =/= 
.0.  /\  ( (coe1 `  f ) `  ( D `  f )
)  =  .1.  ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ` cfv 5255   Basecbs 13148   0gc0g 13400   1rcur 15339  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  Monic1pcmn1 19511
This theorem is referenced by:  ismon1p  19528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-mon1 19516
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